## walters Group Title Question based on (Groups) one year ago one year ago

1. walters Group Title

jjj let the mapping $\tau _{ab}$ for a,b element R ,maps the reals by the ruel $\tau _{ab}:x \rightarrow ax+b.Let G={\tau _{ab}:a \neq 0}$ Determine whether or not G is an abelian group under the composition of mappings

2. walters Group Title

oops ignore jjj

3. KingGeorge Group Title

Well, let $$\tau_{ab},\tau_{cd}\in G$$. Then $\tau_{ab}\circ\tau_{cd}(x)=\tau_{ab}(cx+d)=acx+ad+b.$Also, $\tau_{cd}\circ\tau_{ab}(x)=\tau_{cd}(ax+b)=cax+bc+d.$Since $$acx+ad+b\neq acx+bc+d$$, it must be that $$G$$ is not abelian.

4. walters Group Title

am i not suppose to check whether is a group using the condition before i can conclude that is an abelian group

5. KingGeorge Group Title

It just asked if it was an abelian group. Since the elements don't commute, it can't be an abelian group even if it is a group. However, if we want to check it's a group, we can do that as well. Have you shown any of the axioms for proving it's a group yet?

6. walters Group Title

i am failing to show the axioms

7. KingGeorge Group Title

First, we can see that $$G$$ is closed under composition. $\tau_{ab}\circ\tau_{cd}(x)=\tau_{ab}(cx+d)=acx+ad+b=\tau_{(ac)(ad+b)}.$We also have the identity $$\tau_{10}$$. Now we just have to check associativity and inverses.

8. walters Group Title

why do u chose to use |dw:1360875075213:dw|

9. KingGeorge Group Title

"$$\circ$$" is a relatively common notation to denote the composition of two functions. You can also use $$\tau_{ab}(\tau_{cd}(x))$$ is you prefer.

10. walters Group Title

can u also use * if u wnt

11. KingGeorge Group Title

For inverses, we want $\tau_{ab}\circ\tau_{cd}(x)=acx+ad+b=x$for some $$c,d\in\mathbb{R}$$. We can see that $$c=a^{-1}$$ and $$d=a^{-1}(-b)$$ work for this direction. For the other, $\tau_{cd}\circ\tau_{ab}(x)=cax+bc+d=a^{-1}ax+ba^{-1}-a^{-1}b=x,$so we do indeed have inverses.

12. KingGeorge Group Title

If you make it clear your group function is composition of functions, you can use * as well.

13. KingGeorge Group Title

As for associativity, I'm just going to use the fact that composition of functions is always associative. If you have not heard of that yet, we can still go over how to prove this particular function is associative.

14. walters Group Title

15. KingGeorge Group Title

let $$\tau_{ab},\tau_{cd},\tau_{ef}\in G$$. Then$\tau_{ab}\circ(\tau_{cd}\circ\tau_{ef})(x)=\tau_{ab}=\tau_{ab}(cex+cf+d)=acex+acf+ad+b$and$(\tau_{ab}\circ\tau_{cd})\circ\tau_{ef}(x)=(\tau_{(ac)(ad+b)})(ex+f)=acex+acf+ad+b$Since these are equal, it's associative, and we have a group.

16. walters Group Title

k it is a group but not an abelian group

17. KingGeorge Group Title

Yup.

18. walters Group Title

thnx

19. KingGeorge Group Title

You're welcome.

20. walters Group Title

can we verify identity

21. KingGeorge Group Title

$\tau_{10}(x)=1x+0=x$